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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds   <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/>



<a NAME="E1:22"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E2:22"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T7" title="WELLORD2:th.7">WELLORD2:7</a></i>;<br/>
<a NAME="E3:22"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#NR3" title="ORDERS_1:NR.3">is_linear-order</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T107" title="ORDERS_1:th.107">ORDERS_1:107</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">fab</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:22"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c4">fab</font>
 <b>and </b><br/><a NAME="E6:22"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c4">fab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#D4" target="_self" title="BAGORDER:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">fa</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:22"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c5">fa</font>
 <b>and </b><br/><a NAME="E9:22"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#D4" target="_self" title="BAGORDER:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">fb</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:22"/><i><font color="Green" title="E22">A7</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c6">fb</font>
 <b>and </b><br/><a NAME="E12:22"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c6">fb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#D4" target="_self" title="BAGORDER:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">fab'</font> = <font color="Maroon" title="c4">fab</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">sasb</font> = <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">sasb</font> = <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<b>set </b><font color="Maroon" title="c10">s</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font>;<br/>
<b>set </b><font color="Maroon" title="c11">fa'b</font> = <font color="Maroon" title="c2">a</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">fb'a</font> = <font color="Maroon" title="c3">b</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font></span>)</span>;<br/>
<a NAME="E15:22"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c9">sasb</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="bagorder.html#E3:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<a NAME="E16:22"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">sasb</font>
 
<b>by </b><i><a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E17:22"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">sasb</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T42" title="POLYNOM1:th.42">POLYNOM1:42</a></i>;<br/>
<a NAME="E18:22"/><i><font color="Green" title="E26">A11</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">fa'b</font> = <font color="Maroon" title="c2">a</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font></span>)</span>, <font color="Maroon" title="c14">fb'a</font> = <font color="Maroon" title="c3">b</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="bagorder.html#E16:22"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></i>;<br/>
<a NAME="E20:22"/><i><font color="Green" title="E27">A12</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">fa'b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">fb'a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<a NAME="E21:22"/><i><font color="Green" title="E28">A13</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">fa'b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">fb'a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="valued_0.html#D6" title="VALUED_0:def.6">VALUED_0:def 6</a></i>;<br/>
<a NAME="E22:22"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">fa'b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">fb'a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="bagorder.html#E20:22"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">fa'b</font> = <font color="Maroon" title="c13">fa'b</font>, <font color="Maroon" title="c16">fb'a</font> = <font color="Maroon" title="c14">fb'a</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">fa'bn</font> = <font color="Maroon" title="c15">fa'b</font>, <font color="Maroon" title="c18">fb'an</font> = <font color="Maroon" title="c16">fb'a</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="bagorder.html#E21:22"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">ln</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font>;<br/>
<a NAME="E25:22"/><i><font color="Green" title="E29">A14</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E30">A15</font></i>:  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">fab</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c9">sasb</font></span>)</span>
<b>by </b><i><a class="txt" href="bagorder.html#E6:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="bagorder.html#E16:22"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="bagorder.html#E17:22"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="bagorder.html#E25:22"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
;<br/>
<a NAME="E27:22"/><b>then </b><i><font color="Green" title="E31">A16</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">fa'b</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E16:22"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="bagorder.html#E18:22"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E28:22"/><i><font color="Green" title="E32">A17</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">fb'a</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E16:22"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="bagorder.html#E18:22"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="bagorder.html#E26:22"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E29:22"/><i><font color="Green" title="E33">A18</font></i>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c5">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">fa'bn</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E9:22"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="bagorder.html#T13" target="_self" title="BAGORDER:th.13">Th13</a></i>;<br/>
<a NAME="E30:22"/><i><font color="Green" title="E34">A19</font></i>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c6">fb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c18">fb'an</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E12:22"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="bagorder.html#T13" target="_self" title="BAGORDER:th.13">Th13</a></i>;<br/>
<a NAME="E31:22"/><i><font color="Green" title="E35">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">fa'b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">fb'a</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E27:22"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="bagorder.html#E28:22"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E32:22"/><b>then </b><i><font color="Green" title="E36">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">fa'b</font>
 
<b>by </b><i><a class="ref" href="integra5.html#T2" title="INTEGRA5:th.2">INTEGRA5:2</a></i>;<br/>
<a NAME="E33:22"/><b>then </b><i><font color="Green" title="E37">A22</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E27:22"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">fa'b'</font> = <font color="Maroon" title="c15">fa'b</font> as  <a href="seqm_3.html#NV7" title="SEQM_3:NV.7">natural-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <b>by </b><i><a class="txt" href="bagorder.html#E27:22"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:22_2"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fab'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> holds <br/><font color="Maroon" title="c7">fab'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> ) )</span><br/><b>thus </b><a NAME="E2:22_2"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E27:22"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="bagorder.html#E32:22"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> holds <br/><font color="Maroon" title="c7">fab'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span><br/><b>let </b><font color="Maroon" title="c21">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span> implies <font color="Maroon" title="c7">fab'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">k</font> )</span><br/><b>assume </b><a NAME="E3:22_2"/><i><font color="Green" title="E38">A23</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fab</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">fab'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span><br/><b>reconsider </b><font color="Maroon" title="c22">k1</font> = <font color="Maroon" title="c21">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">fa'bk</font> = <font color="Maroon" title="c15">fa'b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">k1</font>, <font color="Maroon" title="c24">fb'ak</font> = <font color="Maroon" title="c16">fb'a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">k1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>thus </b><font color="Maroon" title="c7">fab'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span>
<b>by </b><i><a class="txt" href="bagorder.html#E6:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="bagorder.html#E17:22"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="bagorder.html#E26:22"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="bagorder.html#E3:22_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">fa'b'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="bagorder.html#E17:22"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="bagorder.html#E26:22"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="bagorder.html#E3:22_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c23">fa'bk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">fb'ak</font>
<b>by </b><i><a class="txt" href="bagorder.html#E17:22"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="bagorder.html#E26:22"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="bagorder.html#E3:22_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
<b>by </b><i><a class="txt" href="bagorder.html#E33:22"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="bagorder.html#E3:22_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E36:22"/><b>then </b>
<font color="Maroon" title="c7">fab'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">fa'b</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c16">fb'a</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<b>hence </b><a NAME="E37:22"/>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="bagorder.html#E8:22"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="bagorder.html#E11:22"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="bagorder.html#E29:22"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="bagorder.html#E30:22"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="bagorder.html#E31:22"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="integra5.html#T2" title="INTEGRA5:th.2">INTEGRA5:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
